when \(\mathcal {G}\)  is strongly connected, where \(= [x_1^T, \cdots , x_n^T]^T\) . Let \(x^*\)  be an optimal solution of cenp, then \(^* = _n\otimes x^*\)  is an optimal solution of disp, vice versa. Consequently, we can solve cenp by equivalently solving disp via the well-known distributed gradient method [1]}:
 
 = k - (k)f(k),

where \(\alpha (k)>0\)  is the time-variant step-size, \(= W\otimes I_m\) , and \(W \in \mathbb {R}^{n\times n}\)  is a weight matrix associated with \(\mathcal {G}\) . The convergence of dgd can be guaranteed with some assumptions about \(\alpha (k)\)  and \(W\) , one of whom is \(\lim _{k\rightarrow \infty }\alpha (k)\rightarrow 0\) . In fact, dgd can not converge exactly to \(^*\)  without the above condition (for example, let \(\alpha (k)\)  be a constant), since \(\nabla f_i(x^*) \ne 0\)  in general. Obviously the diminishing step-size \(\alpha (k)\)  will slow down the convergence rate greatly, hence there is a dilemma: using a constant step-size, the convergence rate is fast but the exact convergence can not be guaranteed; using a diminish step-size, the exact convergence can be guaranteed but the convergence rate is slow. To get rid of the dilemma, we can resort to the discrete-time dynamic average consensus (DAC) algorithm [2]}
 
 = k + f() - f(k)

to track the global gradient. Along gt, theoretically, \(^k\)  will converge to the global gradient \(\frac{1}{n}\sum _{i=1}^n\nabla f_i(x_i^k)\) , hence \(^k\)  can be seen as the inexact global gradient. Based on the above gradient tracking approach, many algorithms with constant step-sizes have been proposed [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, one of whom is DIGing [4]}, [3]}, a Combine-then-Adapt (CTA) algorithm written as
 
 = k - k,
